
considering \(({{\beta }}\mathcal {B}_{2\mathbb {N}}(M),{{\beta }}\mathcal {B}_{2\mathbb {N}+1}(M))\) , then

cancelling (with multiplicity) all the bars that appear in both \({{\beta }}\mathcal {B}_{2\mathbb {N}}(M)\)  and \({{\beta }}\mathcal {B}_{2\mathbb {N}+1}(M)\) , to obtain a signed barcode \((\mathcal {B},\) , and finally

constructing the Hilbert decomposition \((\bigoplus _{i \in \mathcal {B}} F_i, \bigoplus _{j \in  F_j).}This Hilbert decomposition is minimal by construction.\) In order to prove uniqueness, note that, given \((P^*,Q^*)\)  and \((P,Q)\)  as in the statement, we have \(\mathsf {Hil}(P^*) - \mathsf {Hil}(Q^*) = \mathsf {Hil}(P) - \mathsf {Hil}(Q)\) , which implies
\(\mathsf {Hil}(P^* \oplus Q) = \mathsf {Hil}(P^*) + \mathsf {Hil}(Q) = \mathsf {Hil}(P) + \mathsf {Hil}(Q^*) = \mathsf {Hil}(P\oplus Q^*),\) 
and thus \(P^* \oplus Q \cong P \oplus Q^*\)  by hilbert-complete-free.
It follows that \(\mathcal {B}(P^*) \cup \mathcal {B}(Q) = \mathcal {B}(P) \cup \mathcal {B}(Q^*)\) , and so that \(\mathcal {B}(P^*) = \mathcal {B}(P)\)  and \(\mathcal {B}(Q^*) = \mathcal {B}(Q)\)  by minimality.
The argument in the existence part of the proof of existence-minimal-hilbert shows that any Hilbert decomposition can be made minimal by cancelling (with multiplicity) the summands corresponding to bars that appear in the positive and in the negative parts.
An interesting consequence of this is that, for finitely presentable \(M\)  and \(N\) , we have \(\widehat{d_B}({{\beta }}\mathcal {B}(M),{{\beta }}\mathcal {B}(N)) = 0\)  if and only if \(\mathsf {Hil}(M) = \mathsf {Hil}(N)\) .
Recall that \({\mathcal {HB}}(M) = \left(\mathcal {B}(P^*), \mathcal {B}(Q^*)\right)\)  denotes the signed barcode coming from a minimal Hilbert decomposition \((P^*, Q^*)\)  of \(\mathsf {Hil}(M)\) .
The following example implies that there is no constant \(c \geqslant 1\)  such that \(\widehat{d_B}({\mathcal {HB}}(M), {\mathcal {HB}}(N)) \leqslant \, c\cdot d_I(M,N)\) .
<FIGURE>For \(i < j \in \mathbf {R}^n\) , let \(L_{i,j} : \mathbf {R}^n \rightarrow \mathbf {vec}\)  denote the interval module with support \(\lbrace x \in \mathbf {R}^n : i \leqslant x, j x\rbrace \)  (see instability-minimal-decomposition-figure for an illustration).
Note that there is a minimal free resolution \(0 \rightarrow F_j \rightarrow F_i \rightarrow L_{i,j}\) , so that \({{\beta }}\mathcal {B}(L_{i,j}) = (\lbrace i\rbrace ,\lbrace j\rbrace )\) .
Let \(\varepsilon > 0\)  and \(n \in \mathbb {N}\) , and consider the module
\(M = L_{(0,0),(\varepsilon ,\varepsilon )} \oplus L_{(\varepsilon ,\varepsilon ),(2\varepsilon ,2\varepsilon )} \oplus \cdots \oplus L_{((n-1)\varepsilon ,(n-1)\varepsilon ),(n\varepsilon ,n\varepsilon )}.\) 
The module \(M\)  is \(\varepsilon /2\) -interleaved with the zero module 0.
But \({\mathcal {HB}}(M) = \left(\lbrace (0,0)\rbrace ,\lbrace (n\varepsilon ,n\varepsilon )\rbrace \right)\) , so the optimal matching between \({\mathcal {HB}}(M)\)  and \({\mathcal {HB}}(0) = (\emptyset , \emptyset )\)  has cost \(n\varepsilon \) , which can be made arbitrarily large by increasing \(n\) . Thus, by cancelling out all the bars that are common to the positive and negative parts of the signed barcode of \(M\) , the minimal Hilbert decomposition makes it impossible to build low-cost matchings with signed barcodes coming from certain nearby persistence modules like the zero module. By contrast, the Betti signed barcode keeps enough common bars (typically the two copies of each point \((k\varepsilon , k\varepsilon )\)  for \(k=1\)  to \(n-1\)  in this example) to allow for low-cost matchings, according to main-theorem.
1-Wasserstein stability of minimal Hilbert decompositions
In this section, we prove wasserstein-stability.
The section is structured as follows.
In signed-wasserstein-distance-subsection we define the signed 1-Wasserstein distance \(\widehat{d_W}\)  and establish some of its main properties.
In presentation-distance-subsection we recall the definition of the 1-presentation distance \({d_I^1}\)  introduced by Bjerkevik and Lesnick in [1]}.
In proof-of-wasserstein-stability-subsection we recall necessary background from [1]} and prove wasserstein-stability.
The key results from [1]} that we use are recalled below as useful-universality-lp and grid-lemma.
The signed 1-Wasserstein distance
Let \(\mathcal {B}\)  and \( be finite barcodes.Let \) h : B be a bijection of multisets of elements of \(\mathbf {R}^n\) .
The 1-Wasserstein cost of \(h\)  is \(\mathsf {cost}(h) = \sum _{i \in \mathcal {B}} \Vert i - h(i)\Vert _1\) .
Define the following extended pseudodistance on finite (unsigned) barcodes
\(d_W(\mathcal {B}, = \inf \big \lbrace \,\varepsilon \geqslant 0 \,: \,\exists \text{ bijection }\, h : \mathcal {B}\rightarrow { with } \mathsf {cost}(h)\leqslant \varepsilon \,\big \rbrace  \in \mathbb {R}_{\geqslant 0}\cup \lbrace \infty \rbrace .\) 
The signed 1-Wasserstein distance, a dissimilarity function on finite signed barcodes, is given by
\(\widehat{d_W}(\mathcal {B}, = d_W\big (\mathcal {B}_+ \cup -,\, + \cup \mathcal {B}_-\big ),\) 
for all signed barcodes \(\mathcal {B}= (\mathcal {B}_+,\mathcal {B}_-)\)  and \( (+,-)\) .
A finite signed barcode \((\mathcal {B}_+,\mathcal {B}_-)\)  is reduced if \(\mathcal {B}_+\)  and \(\mathcal {B}_-\)  are disjoint.
By cancelling bars with multiplicity, it is easy to see that, for every finite signed barcode \(\mathcal {B}= (\mathcal {B}_+,\mathcal {B}_-)\) , there exists a reduced signed barcode \(\overline{\mathcal {B}} = (\overline{\mathcal {B}}_+, \overline{\mathcal {B}}_-)\)  which is maximal with respect to inclusion and which satisfies \(\overline{\mathcal {B}}_+ \subseteq \mathcal {B}_+\)  and \(\overline{\mathcal {B}}_- \subseteq \mathcal {B}_-\) .
The signed barcode \(\overline{\mathcal {B}}\)  is unique up to equality of multisets.
Note that, with this definition, we have \(\overline{{{\beta }}\mathcal {B}(M)} = {\mathcal {HB}}(M)\)  for any finitely presented persistence module \(M\colon \mathbf {R}^n \rightarrow \mathbf {vec}\) .

For any pair of finite signed barcodes \(\mathcal {B}\)  and \(, we have \) dW(B, = 0\( if and only if \)B=\(.\item The dissimilarity \) dW\( satisfies the triangle inequality.\item For any pair of finite signed barcodes \) B\( and \) , we have \(\widehat{d_W}(\mathcal {B}, = \widehat{d_W}(\overline{\mathcal {B}},\overline{).}\) 

We start with the first claim.
Let \(\mathcal {B}\)  and \( be finite signed barcodes.It is clear that \)B = \( implies \) dW(B, = 0\(.For the converse, we may assume given a bijection \) h : B+ - + B-\( with \) cost(h) = 0\(.Let \) B'+ B+\( be the multiset of elements of \) Rn\( given by the elements of \) B+\( that \) h\( maps to some element of \) +\(.Similarly, define \) - -\( as the multiset of elements that \) h\( maps to \) B-\(.Analogously, define \) +\( and \) B'-\( using the inverse of the bijection \) h\(.It is then clear that \) h\( restricts to bijections \) B'+ +\( and \) - B'-\( of cost \) 0\(.Finally, note that \)(B'+,B'-) = B\( and \)(+,-) = \(.\) For the second claim, let \((\mathcal {B}_+,\mathcal {B}_-)\) , \((+,-)\) , and \((+,-)\)  be finite signed barcodes, and let \(h : \mathcal {B}_+ \cup - \rightarrow + \cup \mathcal {B}_-\)  and \(g : + \cup - \rightarrow + \cup -\)  be bijections with \(\mathsf {cost}(h) = \varepsilon \)  and \(\mathsf {cost}(g) = \delta \) .
It is enough to prove that there exists a bijection \(\mathcal {B}_+ \cup - \rightarrow + \cup \mathcal {B}_-\)  of cost at most \(\varepsilon + \delta \) .
Such a bijection can be defined constructively, as follows.
Given any \(i \in \mathcal {B}_+ \cup - \cup + \cup -\) , let
\(s(i) = {\left\lbrace \begin{array}{ll}h(i), &\text{ if $i \in \mathcal {B}_+ \cup -$},\\g(i), &\text{ if $i \in + \cup -$}.\end{array}\right.}\) 
We claim that, starting from \(i \in \mathcal {B}_+ \cup -\) , one can apply \(s\)  repeatedly until, after finitely many applications, one gets an element \(s^\ast (i) \in + \cup \mathcal {B}_-\) , and that \(s^\ast : \mathcal {B}_+ \cup - \rightarrow + \cup \mathcal {B}_-\)  is a bijection of cost at most \(\varepsilon + \delta \) .
In order to see this, consider the directed graph \(G\)  with vertices given by \(\mathcal {B}_+ \cup \mathcal {B}_- \cup + \cup - \cup + \cup -\)  and a directed edge \(i \rightarrow j\)  if and only if \(j = s(i)\) .
Note that the vertices in \(\mathcal {B}_+ \cup -\)  have in-degree 0 and out-degree 1, the vertices in \(\mathcal {B}_- \cup +\)  have in-degree 1 and out-degree 0, and the vertices in \(+ \cup -\)  have in-degree 1 and out-degree 1.
This implies that \(s^\ast \)  is well-defined and injective.
An analogous argument using the inverses of \(h\)  and \(g\)  shows that \(s^\ast \)  is surjective, and thus a bijection.
Finally, the cost of \(s^\ast \)  is at most \(\varepsilon + \delta \)  since each edge of \(G\)  belongs to at most one path from an \(i \in \mathcal {B}_+ \cup -\)  to \(s^\ast (i) \in + \cup \mathcal {B}_-\) .
The third claim follows from the triangle inequality of \(\widehat{d_W}\)  (second claim) and the fact that \(\widehat{d_W}(\mathcal {B},\overline{\mathcal {B}}) = 0\)  for any finite signed barcode \(\mathcal {B}\)  (first claim).
It is worth pointing out why the proof of the triangle inequality for the signed 1-Wasserstein distance does not work in the case of the bottleneck dissimilarity.
Suppose one follows an argument analogous to the one in the proof of properties-signed-wass(2.), using \(\widehat{d_B}\)  instead of \(\widehat{d_W}\) . With the notation of the proof, given an \(\varepsilon \) -bijection \(h : \mathcal {B}_+ \cup - \rightarrow + \cup \mathcal {B}_-\)  and a \(\delta \) -bijection \(g : + \cup - \rightarrow + \cup -\) , one constructs a bijection \(s^\ast : \mathcal {B}_+ \cup - \rightarrow + \cup \mathcal {B}_-\) .
The problem one encounters is that \(s^\ast \)  may not be an \((\varepsilon +\delta )\) -bijection, for the following reason.
Given \(i \in \mathcal {B}_+\) , the directed path from \(i\)  to \(s^\ast (i) \in + \cup \mathcal {B}_-\)  in the graph \(G\)  contains an even number of edges, with the even edges of bottleneck cost at most \(\varepsilon \)  and the odd edges of bottleneck cost at most \(\delta \) .
But, the path may contain more than two edges, possibly resulting in a bottleneck cost strictly greater than \(\varepsilon + \delta \) .
An analogous problem arises when \(i \in -\) .
This is exactly what happens in cases such as the one of possible-issue-example.
Let \(n \in \mathbb {N}\) , and let \(\mathbf {pmod}\)  denote the set of isomorphism classes of finitely presentable \(n\) -parameter persistence modules.
Define \(\mathbf {Hils}= \lbrace \mathsf {Hil}(M) : M \in \mathbf {pmod}\rbrace \) , the set of all Hilbert functions of finitely presentable \(n\) -parameter persistence modules.
Given \(\eta \in \mathbf {Hils}\) , let \(M \in \mathbf {pmod}\)  be such that \(\eta = \mathsf {Hil}(M)\) .
Note that the signed barcode \({\mathcal {HB}}(M)\)  only depends on \(\eta \)  and not on the specific choice of \(M\) .
In particular, we can define an extended distance \(d_W^\mathbf {Hils}\)  on \(\mathbf {Hils}\)  by letting, for all \(\eta ,\theta \in \mathbf {Hils}\) ,
\(d_W^\mathbf {Hils}(\eta ,\theta ) = \widehat{d_W}({\mathcal {HB}}(M),{\mathcal {HB}}(N)),\) 
where \(M\)  and \(N\)  are any two persistence modules such that \(\eta = \mathsf {Hil}(M)\)  and \(\theta = \mathsf {Hil}(N)\) .
With hilbert-stability-remark in mind, wasserstein-stability can be interpreted as a stability result for Hilbert functions, which we state as the following corollary.
For finitely presentable 2-parameter persistence modules \(M\)  and \(N\)  we have
\(d_W^\mathbf {Hils}(\mathsf {Hil}(M),\mathsf {Hil}(N)) \leqslant 2\, {d_I^1}(M,N).\hfill \) 
The presentation distance
A presentation matrix \(P\)  consists of numbers \(r,c \in \mathbb {N}\) , a matrix \(\mu \in \mathbb {k}^{r \times c}\) , and a function \(\mathcal {L}(P) : \lbrace 1, \cdots , r + c\rbrace  \rightarrow \mathbf {R}^n\)  such that, if the matrix coefficient \(\mu _{i,j} \ne 0\) , then \(\mathcal {L}(P)(i) \leqslant \mathcal {L}(P)(r+j)\) .
We refer to \(\mu \)  as the underlying matrix of \(P\) .
We say that \(P\)  is a presentation matrix of a persistence module \(M : \mathbf {R}^n \rightarrow \mathbf {vec}\)  if \(M\)  is isomorphic to the cokernel of the morphism
\(\bigoplus _{r+1 \leqslant j \leqslant r+c} F_{\mathcal {L}(P_M)(j)} \rightarrow \bigoplus _{1 \leqslant i \leqslant r} F_{\mathcal {L}(P_M)(i)}\)  that has \(\mu \)  as its matrix of coefficients.
Let \(M,N : \mathbf {R}^n \rightarrow \mathbf {vec}\)  be finitely presentable.
Denote by \(\mathcal {P}_{M,N}\)  be the set of all pairs \((P_M,P_N)\)  consisting of presentation matrices \(P_M\)  and \(P_N\)  of \(M\)  and \(N\)  respectively, such that \(P_M\)  and \(P_N\)  have the same underlying matrix.
Define a dissimilarity \(\overline{{d_I^1}}\)  on finitely presentable \(n\) -parameter persistence modules by
\(\overline{{d_I^1}}(M,N) = \inf _{(P_M,P_N) \in \mathcal {P}_{M,N}} \Vert \mathcal {L}(P_M) - \mathcal {L}(P_N)\Vert _1 = \inf _{(P_M,P_N) \in \mathcal {P}_{M,N}} \sum _{i = 1}^{r+c} \Vert \mathcal {L}(P_M)(i) - \mathcal {L}(P_N)(i)\Vert _1.\) 
The following definition of the 1-presentation distance is equivalent to the original definition of Bjerkevik and Lesnick.
[cf. [1]}]

The 1-presentation distance \({d_I^1}\)  is the largest extended pseudodistance on finitely presentable \(n\) -parameter persistence modules that is bounded above by \(\overline{{d_I^1}}\) .

Proof of 1-Wasserstein stability of Hilbert functions
Let \(S\)  be a set and let \(f,g : S \rightarrow \mathbf {R}\)  be functions.
We say that \(g\)  is \(f\) -compatible if \(f(x) \leqslant f(y)\)  implies \(g(x) \leqslant g(y)\) .
If \(f,g : S \rightarrow \mathbf {R}^n\) , we say that \(g\)  is \(f\) -compatible if, for every \(1 \leqslant k \leqslant n\) , the function \(g_k : S \rightarrow \mathbf {R}\)  given by \(g_k(s) = g(s)_k\)  is \(f_k\) -compatible.
The following result and its proof are implicit in the proof of [1]}.
The result gives a useful recipe for proving stability with respect to the presentation distance.
Let \(d\)  be an extended pseudodistance on finitely presentable \(n\) -parameter persistence modules.
In order to prove \(d \leqslant {d_I^1}\) , it is enough to prove the following:

Let \(P_M\)  and \(P_N\)  be presentation matrices of finitely presentable \(n\) -parameter persistence modules \(M\)  and \(N\)  with the same underlying matrix \(\mu \in \mathbb {k}^{r\times c}\)  and such that \(\mathcal {L}(P_N)\)  is \(\mathcal {L}(P_M)\) -compatible.
Then \(d(M,N) \leqslant \Vert \mathcal {L}(P_M) - \mathcal {L}(P_N)\Vert _1\) .

Let \(n \in \mathbb {N}\) .
A grid function is a function \(\mathcal {X}: \mathbf {R}^n \rightarrow \mathbf {R}^n\)  of the form \(\mathcal {X}(a_1, \cdots , a_n) = (\mathcal {X}_1(a_1), \cdots , \mathcal {X}_n(a_n))\)  such that, for all \(1 \leqslant k \leqslant n\) , we have that \(\mathcal {X}_k : \mathbf {R}\rightarrow \mathbf {R}\)  is order preserving, left continuous and satisfies \(\lim _{a \rightarrow \pm \infty } \mathcal {X}_k(a) = \pm \infty \) .
Given a grid function \(\mathcal {X}: \mathbf {R}^n \rightarrow \mathbf {R}^n\) , define \(\mathcal {X}^{-1}(a_1,\cdots ,a_n) = (\mathcal {X}^{-1}_1(a_1),\cdots ,\mathcal {X}^{-1}_n(a_n))\) , where, for \(1 \leqslant k \leqslant n\) , we let
\(\mathcal {X}^{-1}_k(a) = \max \lbrace t \in \mathbf {R}: \mathcal {X}_k(t) \leqslant a\rbrace .\) 
Given a grid function \(\mathcal {X}: \mathbf {R}^n \rightarrow \mathbf {R}^n\)  and a module \(M : \mathbf {R}^n \rightarrow \mathbf {vec}\) , define \(E_\mathcal {X}(M) = M \circ \mathcal {X}^{-1}\) .
Similarly, if \(f : M \rightarrow N\)  is a morphism of persistence modules, one defines \(E_\mathcal {X}(f) : E_\mathcal {X}(M) \rightarrow E_\mathcal {X}(N)\)  using the functoriality of precomposition.
An ordered basis of a free \(n\) -parameter persistence module \(M\)  consists of an ordered list \(B = \lbrace b_1, \cdots , b_k\rbrace \)  of elements of \(\mathbf {R}^n\)  such that \(M\)  is isomorphic to \(\bigoplus _{1 \leqslant i \leqslant k} F_{b_i}\) .
If \(\mathcal {X}: \mathbf {R}^n \rightarrow \mathbf {R}^n\)  is a grid function and \(B = \lbrace b_1, \cdots , b_k\rbrace \)  is an ordered basis of a module \(M\) , define \(E_\mathcal {X}(B) = \lbrace \mathcal {X}^{-1}(b_1), \cdots , \mathcal {X}^{-1}(b_k)\rbrace \) , which is an ordered basis of \(E_\mathcal {X}(M)\) .
If \(B = \lbrace b_1, \cdots , b_k\rbrace \)  and \(B^{\prime }= \lbrace b^{\prime }_1, \cdots , b^{\prime }_k\rbrace \)  are ordered bases of the same length \(k\) , we let \(\Vert B - B^{\prime }\Vert _1 = \sum _{1 \leqslant i \leqslant k} \Vert b_i - b^{\prime }_i\Vert _1\) .
[[1]}]

Suppose given a morphism of finitely generated free 2-parameter persistence modules \(f : P \rightarrow Q\) , a grid function \(\mathcal {X}: \mathbf {R}^2 \rightarrow \mathbf {R}^2\) , and ordered bases \(B\)  and \(C\)  of \(P\)  and \(\mathsf {ker}(f)\) , respectively.
Let \(B^{\prime } = E_\mathcal {X}(B)\)  and let \(C^{\prime } = E_\mathcal {X}(C)\) .
Then, \(\Vert C - C^{\prime }\Vert _1 \leqslant \Vert B - B^{\prime }\Vert _1\) .
We can now prove the main result of this section.
[Proof of wasserstein-stability]
By useful-universality-lp, we may assume that we have presentations \(P_M\)  and \(P_N\)  for \(M\)  and \(N\)  respectively that have the same underlying matrix, and such that \(\mathcal {L}(P_N)\)  is \(\mathcal {L}(P_M)\) -compatible.
We must show that
\(\widehat{d_W}({\mathcal {HB}}(M),{\mathcal {HB}}(N)) \leqslant 2 \Vert \mathcal {L}(P_M) - \mathcal {L}(P_N)\Vert _1\) .
The presentation \(P_M\)  induces a free resolution
\(0 \rightarrow K_M \rightarrow P \rightarrow Q \rightarrow M\) , where \(P = \bigoplus _{r+1 \leqslant j \leqslant r+c} F_{\mathcal {L}(P_M)(j)}\)  and \(Q = \bigoplus _{1 \leqslant i \leqslant r} F_{\mathcal {L}(P_M)(i)}\) .
The module \(K_M\)  is simply the kernel of the morphism \(P \rightarrow Q\) , which is free by Hilbert's syzygy theorem for \(n=2\) .
Then
\(\widehat{d_W}\Big ({\mathcal {HB}}(M), \big (\mathcal {L}(P_M)(\lbrace 1,\cdots ,r\rbrace ) \cup \mathcal {B}(K_M)\,\,,\,\, \mathcal {L}(P_M)(\lbrace r+1,\cdots ,r+c\rbrace )\big )\Big ) = 0,\) 
by properties-signed-wass(1.) and the fact that \((Q \oplus K_M,P)\)  is a Hilbert decomposition of \(\mathsf {Hil}(M)\) , by the rank-nullity theorem.
By an analogous construction for \(N\)  and the triangle inequality for \(\widehat{d_W}\) , it is enough to show that

dW(L(PM)({1,,r}) B(KM) L(PN)({r+1,,r+c} ,
L(PN)({1,,r}) B(KN) L(PM)({r+1,,r+c}) 2 L(PM) - L(PN)1.

We clearly have

dW(L(PM)({1,,r}) L(PN)({r+1,,r+c} ,
L(PN)({1,,r}) L(PM)({r+1,,r+c}) L(PM) - L(PN)1,

and thus it is sufficient to prove that \(d_W(\mathcal {B}(K_M), \mathcal {B}(K_N)) \leqslant \Vert \mathcal {L}(P_M) - \mathcal {L}(P_N)\Vert _1\) .
Note that, since \(\mathcal {L}(P_N)\)  is \(\mathcal {L}(P_M)\) -compatible, there must exist a grid function \(\mathcal {X}: \mathbf {R}^2 \rightarrow \mathbf {R}^2\)  with the property that \(\mathcal {L}(P_N) = \mathcal {L}(P_M) \circ \mathcal {X}^{-1}\) .
Restricting \(\mathcal {L}(P_M)\)  to \(\lbrace r, \cdots , r+c\rbrace \)  we get a basis of \(P\) , which we denote by \(B\) .
Let \(B^{\prime } = E_\mathcal {X}(B)\) , let \(C\)  be any basis of \(K_M\) , and let \(C^{\prime } = E_\mathcal {X}(C)\) , which is an ordered basis of \(K_N\) .
By grid-lemma we have \(\Vert C - C^{\prime }\Vert _1 \leqslant \Vert B - B^{\prime }\Vert _1\) .
To conclude the proof, one combines this last inequality with the inequalities \(\Vert B - B^{\prime }\Vert _1 \leqslant \Vert \mathcal {L}(P_M) - \mathcal {L}(P_N)\Vert _1\)  and \(d_W(\mathcal {B}(K_M), \mathcal {B}(K_N)) \leqslant \Vert C - C^{\prime }\Vert _1\) , which hold by construction.

Discussion
Hilbert function versus rank invariant.
As mentioned in the introduction, our work is inspired by the work of Botnan et al. [7]} on signed decompositions of the rank invariant.
Indeed, our definition of Hilbert decomposition is analogous to their definition of rank decomposition.
The difference is that, while we decompose the Hilbert function of a persistence module as a \(\mathbb {Z}\) -linear combination of Hilbert functions of free modules, they decompose the rank invariant of a persistence module as a \(\mathbb {Z}\) -linear combination of rank invariants of rectangle modules.
The analogy goes further.
While we rely on free (hence projective) resolutions, and on the usual notion of exactness, they show that rank decompositions are related to a different exact structure on the category of (multiparameter) persistence modules, the rank-exact structure.
There is an analogue of multigraded Betti numbers for this other exact structure, and in on-going work we show that using this notion one can prove an analogue of main-theorem.
The difficulty in extending main-theorem to the rank-exact structure is twofold: first, to bound the global dimension of this exact structure; second, to extend Bjerkevik's stability result [8]} to the class of projectives of this exact structure.
It is interesting to note that the persistent version of Schanuel's lemma for projective resolutions, stability-resolutions, applies in any exact structure for which the shift functors \((-)[\varepsilon ]\)  are exact.
Recent work of Blanchette, Brüstle, and Hanson [9]} considers the general approach of defining invariants of persistence modules indexed by finite posets using exact structures.
In particular, they introduce a family of exact structures—which includes the rank-exact structure for finite posets—and show that these structures have finite global dimension.
However, since their argument depends on the cardinality of the indexing poset, it is not clear to us that their global dimension result can be used to provide a finite upper bound for the global dimension of the rank-exact structure on finitely presentable \(n\) -parameter persistence modules, since \(\mathbf {R}^n\)  is not a finite poset.
A last point related to [7]} is that an example similar to instability-minimal-decomposition-example shows that minimal rank decompositions in the sense of [7]} are not stable either in the bottleneck dissimilarity.
Tightness of main-theorem. Hilbert's global dimension
bound is tight: \(\mathsf {gl.dim}(\mathbb {k}[x_1,\cdots ,x_n]) = n\) . Meanwhile,
Bjerkevik's bottleneck stability bound for free modules is tight at
least in the cases \(n \in \lbrace 2,4\rbrace \)  [8]}.
Nevertheless, we do not know whether our
bound in main-theorem is tight, even when \(n \in \lbrace 2,4\rbrace \) .
Addressing the tightness of our bound requires understanding exactly
what kinds of matchings can arise from applying Bjerkevik's result on the
interleaving given by stability-resolutions, which is the subject of future work. Note that the bound
\(\widehat{d_B}\big ({{\beta }}\mathcal {B}(M),\, {{\beta }}\mathcal {B}(N)\big )\, \leqslant \, 2\;d_I(M,N)\)  obtained in the case \(n=1\)  is tight: for instance,
\(\widehat{d_B}\big ({{\beta }}\mathcal {B}(\mathbb {k}_{[0,2)}),\, {{\beta }}\mathcal {B}(0)\big )\, =\, 2\) 
while the two modules are 1-interleaved.
Let us also point out that \(\widehat{d_B}\big ({{\beta }}\mathcal {B}(M),\, {{\beta }}\mathcal {B}(N)\big )\)  can be arbitrarily small compared to \(d_I(M,N)\) , even when \(n=1\) , by remark-same-hilbert-function. This is expected of any distance or dissimilarity function on multigraded Betti numbers, since the latter forget about the differentials in the resolutions—hence about the pairing defining the intervals in the persistence barcode in the case \(n=1\) . Take for instance \(M=\mathbb {k}_{[0,2)} \oplus \mathbb {k}_{[1,3)}\)  and \(N=\mathbb {k}_{[0,3)} \oplus \mathbb {k}_{[1,2)}\) , for which we have \({{\beta }}\mathcal {B}(M) = {{\beta }}\mathcal {B}(N)\)  while \(M\lnot \cong N\) , so \(\widehat{d_B}\big ({{\beta }}\mathcal {B}(M),\, {{\beta }}\mathcal {B}(N)\big )=0\)  while \(d_I(M,N) = d_B\big (\mathcal {B}(M),\, \mathcal {B}(N)\big ) = 1> 0\) .
wasserstein-stability for general \(n\) .
The difficulty of extending wasserstein-stability to higher dimensions lies in the fact that, in our proof, we use key results of Bjerkevik and Lesnick that apply to the cases \(n\in \lbrace 1,2\rbrace \) , and for which we currently do not know of any \(n\) -parameter generalizations.
Topologies of the spaces of signed barcodes and of Hilbert functions.
Let \(n \in \mathbb {N}\) , and let \(\mathbf {pmod}\)  denote the set of isomorphism classes of finitely presentable \(n\) -parameter persistence modules.
Although the dissimilarity \(\widehat{d_B}\)  does not satisfy the triangle inequality, it can be used to define a topology on \(\mathbf {sBarc}\) , the set of finite signed barcodes that are the Betti signed barcode of some finitely presentable persistence module.
For the basis of the topology one uses balls as done for defining the topology of a metric space.
With this definition, it follows from main-theorem that the Betti signed barcode operator \({{\beta }}\mathcal {B}: \mathbf {pmod}\rightarrow \mathbf {sBarc}\)  is continuous.
We can then topologize the set \(\mathbf {Hils}\)  of all Hilbert functions of finitely presentable persistence modules, using the final topology induced by the map \(\Sigma : \mathbf {sBarc}\rightarrow \mathbf {Hils}\)  given by
\(\Sigma (\mathcal {B}_+,\mathcal {B}_-) = \mathsf {Hil}\left(\bigoplus _{i \in \mathcal {B}_+} F_i\right) - \mathsf {Hil}\left(\bigoplus _{j \in \mathcal {B}_-} F_j\right)\) .
We refer to this topology on \(\mathbf {Hils}\)  as the bottleneck topology.
Since \(\mathsf {Hil}= \Sigma \circ {{\beta }}\mathcal {B}\) , it follows that \(\mathsf {Hil}: \mathbf {pmod}\rightarrow \mathbf {Hils}\)  is continuous.
The upshot is that \(\mathbf {sBarc}\)  is not \(T_1\) , by remark-same-hilbert-function, whereas \(\mathbf {Hils}\)  endowed with the bottleneck topology is \(T_1\) , since, if \(\mathsf {Hil}(M) \ne \mathsf {Hil}(N)\) , then there is a ball (with respect to \(\widehat{d_B}\) ) around \({{\beta }}\mathcal {B}(M)\)  that does not contain \({{\beta }}\mathcal {B}(N)\) .
In fact, the map \(\Sigma : \mathbf {sBarc}\rightarrow \mathbf {Hils}\)  is the universal map onto a \(T_1\)  space.
Note, however, that the bottleneck topology is not Hausdorff, by possible-issue-example.
Another topology can be defined on \(\mathbf {Hils}\)  by an analogous construction, but using \(\widehat{d_W}\)  instead of \(\widehat{d_B}\) .
We refer to this topology on \(\mathbf {Hils}\)  as the 1-Wasserstein topology.
It is better behaved than the bottleneck topology.
Indeed, the 1-Wasserstein topology is Hausdorff, and in fact it is metrizable, as it can be metrized using the extended distance \(d_W^\mathbf {Hils}\)  discussed in hilbert-stability-remark.
As we do in hilbert-stability-corollary for the case of 2-parameter persistence, one can use \(d_W^\mathbf {Hils}\)  to state a stability result for Hilbert functions.
